Editing Talk:Johnson solid (section)

== duals of the Johnson solids ==

Is there a name for the set of 92 polyhedra that are duals of the Johnson solids?
Other than "duals of the Johnson solids"?
(By analogy with the way [[Catalan solid]]s are duals of the [[Archimedean solid]]s).
--[[User:DavidCary|DavidCary]] ([[User talk:DavidCary|talk]]) 04:10, 6 April 2013 (UTC)

:Not to my knowledge. I don't think they have even been enumerated in any reliable source. I'd probably call them "Johnson duals" for short. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 13:12, 6 April 2013 (UTC)

:: One problem is that for a geometric dual – rather than a mere topological dual – you need a center.  How do you choose centers for the 56 that lack [[List of spherical symmetry groups#Dihedral symmetry|''D'' symmetry]]? (''Ch'' or ''Ci'' or ''S'' symmetry would also do, but there aren't any.)  —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 07:09, 27 February 2014 (UTC)

:::To define the centre usefully, it would need to remain static under duality - that is, the centre of the dual must be the same point as the centre of the original. This ensures that when you dualise the dual, you get back to the original form. It turns out that for some figures this is really hard, I seem to recall that even the "Stella" software author gave up on it and used a simpler algorithm. I think it would be fair to ignore centres and polar reciprocity but instead to require the dual ''condition'', that all vertices be regular, i.e. having the same polygonal angle between adjacent edges. Not sure if that set of polyhedra would match the Johnson solids one-to-one, though: an interesting problem. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 10:37, 27 February 2014 (UTC)